Crooked Timber

I’m sorry the APA is in New York, because it means it is insanely expensive for everyone visiting, especially poor students. If it is going to be in New York, I wish it were in a nicer and cheaper part of town – though I’m not sure how many convention sized hotels there are elsewhere. Still, since it will be in New York, we should talk about where to eat while folks are visiting. I’ve got a lot of suggestions below the fold, and anyone with more suggestions is encouraged to pipe up in comments.

In their Vague Parts and Vague Identity, Elizabeth Barnes and Robbie Williams do some excellent work cleaning up, and noting the missing premises in, an argument of mine against vague parthood. In particular, they note that the argument I run could be improved in two ways.

Let K be an object with s as a vague part. Let K- be the metaphysical difference between K and s, and K+ the metaphysical fusion of K and s. I supposed that K couldn’t be identical to K+ or to K-. They note that this should be argued for, not assumed, and offer an argument.

They note that the Evansian argument I offer to complete the reductio makes a rather strong mereological assumption. It requires not just that for any things, it is determinate that they have a fusion, but that for any things, there is something that determinately is their fusion.

I agree with both points. I hadn’t noticed the potential scope ambiguity that the second observation turns on. Now that I have, I think I accept both disambiguations. But that is certainly a necessary move in the argument, and one that could be rejected.

Given that extra assumption, I think I was intending the argument that K is distinct from K+ and K- in a slightly different way. Here’s how I was thinking of K+ and K-.

K+ is the determinate fusion of K and s.

K- is the determinate mereological difference between K and s.

Now the argument that K is not identical to K+ or K- is a simple application of Leibniz’s Law. Since s is determinately a part of K+, determinately not a part of K-, and vaguely a part of K, it follows immediately that they are not identical.

Note that once we do things this way, we can get a reductio without appeal to any distinctively classical principles, as follows.

Assume, for reductio, that s is part of K.

Then K and K+ coincide.

So K and K+ are identical.

This contradicts our earlier result that K and K+ are not identical.

So s is not part of K.

Hence K and K- coincide.

So K and K- are identical.

This contradicts our earlier result that K and K- are not identical.

So we’ve generated a contradiction from just the assumption that s is a vague part of K, and the definitions of K+ and K-. Since Elizabeth and Robbie don’t believe the strong mereological principles that are used in the definition of K+ and K-, they won’t be overly moved by this reasoning. The target of this argument instead is someone who reads their paper and thinks that dropping classical logic, while accepting the existence of determinate fusions and differences, is consistent with vague identity.

Having said all this, I’m starting to wonder how strong an argument it is against metaphysical vagueness. I’m convinced by the Evans argument against vague identity. And I think a cleaned up version of the above argument shows that we cannot have vague parthood. A similar argument shows that there cannot be vague existence, as follows.

If a vaguely exists, let b be any distinct object, and c be the object that determinately has parts b and, iff it exists, a. Then b and c are distinct, since one but not the other determinately is wholly distinct from a. But if a doesn’t exist, then b and c coincide, and so are identical. Hence a doesn’t exist, and since it was arbitrary, all ‘vaguely existing’ objects don’t exist.

But that doesn’t cover all of the ways we might have metaphysical vagueness. In particular, it doesn’t rule out vague events. None of these arguments rules out the idea that there could be, say, a traffic jam for which it was vague whether a particular car was part of that traffic jam. If we have a plenitudinous ontology of events, then I imagine similar arguments to the ones discussed so far will lead to contradiction. (Let Herbie be the car that’s vaguely in the traffic jam T. Let T- be the event consisting of the traffic jam without Herbie, etc.) But I think the defender of vague events could simply deny that there are principles of event composition that are anywhere near as strong as the principles of classical mereology. (So in the Herbie example, there won’t be an event T-.)

I hope to write more of this in the near future. But for now I just want to note that there’s more to metaphysics than mereology, and even with a precise mereology, we might have some vagueness somewhere in the metaphysics.

This is the final call for papers for the Formal Epistemology Workshop next year in Konstanz. It looks like this year’s FEW will be somewhat smaller than it has been in some years – I remember the second one in particular being quite a marathon, running for something like 8 hours a day for five days! This one looks like it will be a manageable size, but it will still manage to have a lot of great people working in formal epistemology, including many more Europeans than in years when it was in the US.

The submission deadline is December 31, 2009. The whole call for papers follows below.

Stewart Shapiro’s 1998 paper “Induction and indefinite extensibility: The Gödel sentence is true, but did someone change the subject?” makes a really nice point that I hadn’t thought about before. Let’s say that you’re attracted to the following two kinds of claims.

1) Autonomy in logic – A logical truth should be provable using the logical rules for just the connectives in that truth.
2) Logicism in mathematics – Mathematical truths are logically true, in a properly extended sense of ‘logically’.

Several writers, tracing back at least to Dummett, have used (1) to argue against classical logic. The problem is that Pierce’s Law, i.e. ((p → q) → p) → p, is a classical theorem. But, they say, it can’t be proven given just the appropriate logical rules for →. The latter claim is controversial, but plausible.

The problem is that if you don’t like Pierce’s Law for this reason, you should think that ∃x ∃y (x ≠ y) should be provable using just the logical rules for ∃ and ≠, at least if it is a logical truth. Now on the one hand, it doesn’t seem plausible that the rules for ∃ and ≠ suffice to prove ∃x ∃y (x ≠ y). Certainly the intuitionist rules for ∃ and ≠ don’t suffice.

On the other hand, 0 ≠ 1 is a mathematical truth. So ∃x ∃y (x ≠ y) is a mathematic truth. So, given logicism, it is a logical truth. So given autonomy, it should be provable by considerations purely about ∃ and ≠.

Of course, there’s a lot that can be said about various ways out of this puzzle you might try while holding on to something like (1) and (2). But ∃x ∃y (x ≠ y) isn’t the only case where there’s a conflict between (1) and (2), so the ways out have to do a lot of work. Shapiro’s paper is a very nice study of what the options are here, and why they all look fairly problematic. I recommend the paper to anyone who is interested in these issues. (Though, to be fair, there probably aren’t that many people like me who both are interested in these issues and haven’t read the paper already!)

I’ve been talking a lot with Tom Donaldson about various iterations of pragmatism in 20th Century philosophy, especially in America. And I was struck by some things that came up when discussing Rorty.

Rorty sometimes says that the distinctive claim of pragmatism is that we can’t give a positive theory of truth, and should instead settle for noting that ‘true’ can be used disquotationally, and perhaps in a couple of other ways.

Now I think, or at least suspect, we can’t give a positive theory of truth. I’m certainly no expert on the various attempts that have been made to resolve the semantic paradoxes, but I’m sympathetic to views that deny there is a single truth predicate (or operator), and instead say that the best we can do is offer up some kind of Tarskian hierarchy of different truth predicates/operators.

Let’s say I did adopt that Tarskian kind of view. It would then seem that I could agree with a lot of what Rorty says about truth. There is no universal theory of truth. Indeed, it is impossible to even state a full theory of truth, of the kind traditionally desired by philosophers.

But it seems wrong to say that accepting such a view about the paradoxes commits me to being a pragmatist in any broad sense. It seems I’m on quite the opposite side to Rorty about most of the big picture questions. What could have gone wrong?

My guess, and it is really no more than a guess, is that truth is less important to Rorty than inquiry. I think that inquiry (as such) aims at correspondence to reality. I don’t think that’s the same thing as saying it aims at truth, because I don’t hold the correspondence theory of truth. But having a narrowly construed aim of inquiry, and having that aim be defined in correspondence terms, puts me outside the pragmatist camp. This suggestion is meant to be one that both realists and pragmatists can accept; if I’m a Tarskian about truth, and a correspondence theorist about inquiry, then I clearly disagree with them about inquiry, and that seems to be the important disagreement.

I know that I’ve said this before, but given that the issue has come up again, I think it bears repeating.

The standard example people use of something that’s intuitive but not actually true is Frege’s Axiom V. And that axiom, remember, says that any predicate has a set for its extension. Apparently this is intuitive. I don’t find it even approximately intuitive. I’ve been introspecting a bit, and I’ve noticed the following things.

Two days a week, my intuitions are nominalist and so don’t accept any sets, let alone sets for any predicates.

Two days a week, my intuitions have an iterative conception of sets, so there couldn’t possibly be a set of all sets. (How could it get built, since it would have to have itself as a constituent? That’s very counterintuitive.)

Two days a week, my intuitions are modest and say that set theory is too hard for them, and that they aren’t going to issue verdicts about anything to do with set theory.

And on Sundays I watch football.

I don’t doubt that there are some things that are true but counterintuitive. Some of the simple results about comparative advantage in trade theory are counterintuitive but true, for instance. It’s counterintuitive, but I think true, that Mary learns nothing when leaving the black and white room. But I’m very sceptical that Axiom V is one of the counterintuitive truths.

One of the odd things about my preferred logic of vagueness, as set out in True, Truer, Truest is that disjunctions can be perfectly true even though no disjunct is perfectly true. Some people (including me some of the time) find this a little unintuitive. It also has a quirky logical consequence.

I hadn’t been thinking about this at the time, but one consequence of that is that the simplest way to do multiple conclusion logic in the theory will lead to some odd failures of validity. For instance, using ‘:’ as a turnstile, it is natural to think that (1) will fail.

(1) A ∨ B: A, B

If our understanding of validity is that on any interpretation, at least one of the conclusions is at least as true as the conjunction of the premises, then (1) is false. That’s because when B is, say, ¬A, then the premise will be perfectly true, but each disjunct will be less than perfectly true.

Objection: It’s a little odd to have this definition of validity. It requires, in effect, that we treat the premises as a long conjunction, but the conclusions separately. If the rule for validity was that the disjunction of the conclusions must be at least as true as the conjunction of the premises, then (1) would be valid.

Response: I think this objection ultimately doesn’t take the multiple conclusion form of (1) seriously enough. It’s true that you could turn around and say that the theory of validity in True, Truer, Truest doesn’t take the multiple premise form of arguments seriously enough. But to that I happily plead guilty; I’m happy to say that arguments really have one long premise, with the parts sometimes joined by ‘∧’, and sometimes by ‘.’.

I mentioned briefly in the earlier paper that you could do an intuitionist version of the semantics in True, Truer, Truest. The classical version that I developed there said that truer generated a Boolean lattice over (equivalence classes of) propositions (where the classes are of equally true propositions). If instead you said that truer generates a Heyting lattice over (equivalence classes of) propositions, you get an intuitionist version of my theory. (That is, all and only intuitionist theorems are logical truths, and and all and only intuitionistically valid sequents are really valid.)

Now some Heyting lattices have the property that the join of a and b can be the top element even though neither a nor b is the top element. But not all Heyting lattices have this property. In particular, the Lindenbaum-Tarski algebra has this property. That’s the algebra defined over sentences in a countable language ordered by intuitionist entailment. So if we supplement the theory of True, Truer, Truest with the requirement that A ∨ B is perfectly true iff A is perfectly true or B is perfectly true, we get a consistent intuitionist theory.

Having said that, we still don’t make (1) valid. It can still be the case that A ∨ B is strictly weaker than both A and B, and in that case, (1) will be invalid. The trick, I think, will be to change the definition of validity.

New Account of Validity: An argument is valid iff whenever the premises are perfectly true, at least one of the conclusions is perfectly true.

The New Account of Validity doesn’t change what sequents are valid in the classical case, but it does change some validities in the (multiple-conclusion) intuitionist version of the ‘truer’ theory. Note that the New Account doesn’t imply that p entails that p is perfectly true, because p can be perfectly true without it being perfectly true that p is perfectly true.

If we think the algebra of truth values is a Lindenbaum-Tarski algebra and we adopt the New Account of Validity, then (1) will be valid. I’m not sure whether this is too high a price to pay to keep (1) valid, but I think there is some motivation for each of these assumptions.

UPDATE: I fixed the broken link to True, Truer, Truest. Thanks to Seamus Bradley for alerting me to this.

My position on evidence is I think fairly similar to the position Clayton Littlejohn takes when he says evidence is non-inferential knowledge. I think, as I say in this old paper, that evidence is basically knowledge that is the output of a Fodorian module. The differences between our positions aren’t great. But I think there are some differences. This kind of case brings out some of these differences.

Graham, Crispin and Ringo have an audience with the Delphic Oracle, and they are told ¬ p ∨ q and ¬ ¬ p. Graham is a relevant logician, so if he inferred p ∧ q from these pronouncements, his belief in the invalidity of disjunctive syllogism would be a doxastic defeater, and the inference would not constitute knowledge. Crispin is an intuitionist logician, so if he inferred p ∧ q from these pronouncements his belief in the invalidity of double negation elimination would be a doxastic defeater, and the inference would not constitute knowledge. Ringo has no deep views on the nature of logic, but has accepted the classical theory he learned in an undergrad intro class because he doesn’t know there’s any dispute about it. Moreover, in the world of the story classical logic is correct. So if Ringo were to infer p ∧ q from these pronouncements, his belief would constitute knowledge. Now Graham’s and Crispin’s false beliefs about entailment are not p ∧ q-relevant evidence, and Ringo doesn’t have more evidence about logic than Graham or Crispin. So all three of them have the same p ∧ q-relevant evidence, but only Ringo is in a position to know p ∧ q.

This case is meant to do two things. First, it is an argument against a kind of evidentialism about knowledge. It isn’t true that what you know, or even what you’re in a position to know, supervenes on the evidence you have. Graham, Crispin and Ringo have the same evidence, but only Ringo is in a position to know that p ∧ q. That’s because Graham and Crispin’s false beliefs about entailment are defeaters in this context. In general, knowledge doesn’t supervene on evidence because defeaters don’t supervene on evidence.

The other thing it is supposed to do is draw out the idea that there is something problematic about treating logical knowledge as evidence. Ringo knows that ¬ p ∨ q and ¬ ¬ p entail that p ∧ q. Graham and Crispin don’t know this. But this isn’t an extra piece of evidence that Ringo has. Indeed, it isn’t an extra piece of evidence Ringo has for two reasons.

First, Graham and Crispin have all the evidence that Ringo has. They know that logic professors in intro classes say that ¬ p ∨ q and ¬ ¬ p entail that p ∧ q. They just don’t believe that these professors are right. This doesn’t seem like the kind of thing that destroys evidence.

Second, there are reasons tracing back to Lewis Carroll’s “Achilles and the Tortoise” for distinguishing between logical rules and logical axioms. Similar reasons suggest that we should distinguish between between empirical evidence and inferential rules that licence inferences from that evidence. That’s especially true when the inferential rules just are logical rules. So I don’t think that what licences Ringo’s inference of p ∧ q is an extra bit of evidence, it’s rather a rule of logic.

But Ringo knows that rule is correct. Indeed, it might even be a non-inferential piece of knowledge for him. (If need be, make Ringo smart enough in the example that he can simply see that certain inferences are valid.) Basic inferential rules aren’t things we know inferentially, they are things we use to get inferential knowledge. So if evidence is non-inferential knowledge, they are evidence. But I suspect they are not evidence for the reasons I’ve given here. So I think there’s a downside to the equation of evidence with non-inferential knowledge.

Thanks to the feedback I’ve gotten already on evidence. Here are three other questions that I’ve already got from looking at those suggestions, and thinking about related stuff.

1) In the literature on testimony, there are several writers who deny that testimony is a source of evidence. Rather, they say, testimony provides a kind of non-evidential warrant.

Now I’m rather sceptical that this is the right approach to testimony. But one thing that’s worth thinking about from a perspective of a theory of evidence is what it could mean for this to be true. On some theories of evidence (on E=K for instance) it’s hard to see how this could even be a coherent position. I suspect it is a coherent position, and that’s a reason to doubt E=K. But turning this into an argument requires saying more about what non-evidential warrant might be, and hopefully saying that will provide some clues as to what evidential warrant might be!

It might be helpful here to think about the kind of warrant we get from doing simple logical inferences. When we validly infer q from p, and thereby come to know q, presumably p is part of our evidence. But I don’t think it is true in general that the fact that p implies q is part of our evidence. If it is, then we should also say that the fact that p, and p implies q, together imply q is part of the evidence, and so on ad infinitum, and that strikes me as absurd. So in inference there are things that we know, and that provide warrant, but which are not part of our evidence, properly construed. So I am sympathetic to the idea of non-evidential warrant, though I’d like to know a lot more about how it is supposed to work.

2) It’s interesting to think about what evidence a person has. And it’s interesting to think about what evidence a person has for different claims H, where H is a claim that the person could be considering. Here’s a third question: Which of those two concepts is primary? I sometimes think E=K is best considered as a view where the concept of evidence for is primary, and evidence simpliciter is a derived notion.

What I mean by this is to say that E is part of S’s evidence just is to say that E is part of S’s evidence for some hypothesis, and the order of explanation/analysis here goes from right-to-left. One question is whether that’s the right way to take E=K. (I think it makes the best sense of some arguments for E=K.) The other question is whether it is the right way to think about the question more broadly. Should we analyse ‘evidence for’ by first offering a theory of evidence, and then asking what of that evidence is evidence for the hypothesis in question, or should we analyse evidence by first offering a theory of evidence for different hypotheses, and then asking what propositions are evidence for some propositions or other? This seems like a fairly deep question, and I don’t know how to attack it.

3) In law, evidence typically does not consist of propositions. In philosophy we’ve focussed our attention on propopsitional evidence. Or at least we have recently. When Quine said evidence was sensory irritations, he presumably was not saying that our evidence consists of facts about sensory irritations. We might have made a deep mistake here; maybe propositional evidence is just a special kind of evidence.